Method and system for pricing financial derivatives

ABSTRACT

A method for providing a bid price and/or an offer price of an option relating to an underlying asset, the method including the steps of receiving first input data corresponding to a plurality of parameters defining the option, receiving second input data corresponding to a plurality of current market conditions relating to the underlying value, computing a corrected theoretical value (CTV) of the option based on the first and second input data, computing a bid/offer spread of the option based on the first and input data, computing a bid price and/or an offer price of the option based on the corrected TV and the bid/offer spread, and providing an output corresponding to the bid price and/or the offer price of said option.

FIELD OF THE INVENTION

The invention relates generally to financial instruments and, morespecifically, to methods and systems for pricing financial derivativesand for providing automatic trading capabilities.

BACKGROUND OF THE INVENTION

Pricing financial instruments, e.g., financial derivatives, is a complexart requiring substantial expertise and experience. Trading financialinstruments, such as options, involves a sophisticated process ofpricing typically performed by a trader.

The term “option” in the context of the present application is definedbroadly as any financial instrument having option-like properties, e.g.,any financial derivative including an option or an option-likecomponent. This category of financial instruments may include any typeof option or option-like financial instrument, relating to someunderlying asset. Assets as used in this application include anything ofvalue; tangible or non-tangible, financial or non-financial. Forexample, as used herein, options range from a simple Vanilla option on asingle stock and up to complex convertible bonds whose convertibilitydepends on some key, e.g., the weather.

The price of an asset for immediate (e.g., 2 business days) delivery iscalled the spot price. For an asset sold in an option contract, thestrike price is the agreed upon price at which the deal is executed ifthe option is exercised. For example, a foreign exchange (FX) optioninvolves buying or selling an amount of one currency for an amount ofanother currency. The spot price is the current exchange rate betweenthe two currencies on the open market. The strike price is the agreedupon exchange rate of the currency if the option is exercised.

To facilitate trading of options and other financial instruments, atrader prepares a bid price and offer price (also called ask price) fora certain option. The bid price is the price at which the trader iswilling to purchase the option and the offer price is the price at whichthe trader is willing to sell the option. When another trader isinterested in the option the first trader quotes both the bid and offerprices, not knowing whether the second trader is interested in sellingor buying. The offer price is higher than the bid price and thedifference between the offer and bid is referred to as bid-offer spread.

A call option is an option to buy an asset at a certain price (i.e., astrike price) on a certain date. A put option is an option to sell anasset at a strike price on a certain date. At any time prior to theoption expiration date, the holder of the option may determine whetheror not to exercise the option, depending on the current exchange rate(spot) for that currency. If the spot (i.e., current market) price islower than the strike price, the holder may choose not to exercise thecall option and lose only the cost of the option itself. However, if thestrike is lower than the spot, the holder may exercise the right to buythe currency at the strike price making a profit equal to the differencebetween the spot and the strike prices.

A forward rate is the future exchange rate of an asset at a given futureday on which the exchange transaction is performed based on an optioncontract. The forward rate is calculated based on a current rate of theasset, a current interest rate prevailing in the market, expecteddividends (for stocks), cost of carry (for commodities), and otherparameters depending on the underlying asset of the option.

An at-the-money forward option (ATM) is an option whose strike is equalto the forward rate of the asset. In this application, at-the-moneyforward options are generically referred to as at-the-money options, asis the common terminology in the foreign exchange (FX) and otherfinancial markets. An in-the-money call option is a call option whosestrike is below the forward rate of the underlying asset, and anin-the-money put option is a put option whose strike is above theforward rate of the underlying asset. An out-of-the-money call option isa call option whose strike is above the forward rate of the underlyingasset, and an out-of-the-money put option is a put option whose strikeis below the forward rate of the underlying asset.

An exotic option, in the context of this application, is a generic namereferring to any type of option other than a standard Vanilla option.While certain types of exotic options have been extensively andfrequently traded over the years, and are still traded today, othertypes of exotic options had been used in the past but are no longer inuse today. Currently, the most common exotic options include are“barrier” options, “binary” options, “digital” options, “partialbarrier” options (also known as “window” options), “average” options and“quanto” options. Some exotic options can be described as a complexversion of the standard (Vanilla) option. For example, barrier optionsare exotic options where the payoff depends on whether the underlyingasset's price reaches a certain level, hereinafter referred to as“trigger”, during a certain period of time. The “pay off” of an optionis defined as the cash realized by the holder of the option upon itsexpiration. There are generally two types of barrier options, namely, aknock-out option and a knock-in option. A knock-out option is an optionthat terminates if and when the spot reaches the trigger. A knock-inoption comes into existence only when the underlying asset's pricereaches the trigger. It is noted that the combined effect of a knock-outoption with strike K and trigger B and a knock-in option with strike Kand trigger B, both having the same expiration, is equivalent to acorresponding Vanilla option with strike K. Thus, knock-in options canbe priced by pricing corresponding knock-out and vanilla options.Similarly, a one-touch option can be decomposed into two knock-in calloptions and two knock-in put options, a double one-touch option can bedecomposed into two double knock-out options, and so on. It isappreciated that there are many other types of exotic options known inthe art.

Certain types of options, e.g., Vanilla options, are commonlycategorized as either European or American. A European option can beexercised only upon its expiration. An American option can be exercisedat any time after purchase and before expiration. For example, anAmerican Vanilla option has all the properties of the Vanilla optiontype described above, with the additional property that the owner canexercise the option at any time up to and including the option'sexpiration date. As is known in the art, the right to exercise anAmerican option prior to expiration makes American options moreexpensive than corresponding European options. Generally in thisapplication, the term “Vanilla” refers to a European style Vanillaoption. European Vanilla options are the most commonly traded options;they are traded both on exchanges and over the counter (OTC). The muchless common American Vanilla options are traded exclusively OTC, and aredifficult to price.

U.S. Pat. No. 5,557,517 (“the '517 patent”) describes a method ofpricing American Vanilla options for trading in a certain exchange. Thispatent describes a method of pricing Call and Put American Vanillaoptions, where the price of the option depends on a constant margin orcommission required by the market maker. The method of the '517 patentignores data that may affect the price of the option, except for thecurrent price of the underlying asset and, thus, this method can lead toserious errors, for example, an absurd result of a negative optionprice. Clearly, this method does not emulate the way American styleVanilla options are priced in real markets.

The Black-Scholes model (developed in 1975) is a widely accepted methodfor valuing options. This model calculates a probability-basedtheoretical value (TV), which is commonly used as a starting point forapproximating option prices. This model is based on a presumption thatthe change in the rate of the asset generally follows a Brownian motion,as is known in the art. Using such Brownian motion model, known also asa stochastic process, one may calculate the theoretical price of anytype of financial derivative, either analytically, as is the case forthe exotic options discussed above, or numerically. For example, it iscommon to calculate the theoretical price of complicated financialderivatives through simulation techniques, such as the Monte Carlomethod, introduced by Boyle in 1977. Such techniques may be useful incalculating the theoretical value of an option, provided that thecomputer being used is sufficiently powerful to handle all thecalculations involved. In the simulation method, the computer generatesmany propagation paths for the underlying asset, starting at the tradetime and ending at the time of the option expiry. Each path is discreteand generally follows the Brownian motion probability, but may begenerated as densely as necessary by reducing the time lapse betweeneach move of the underlying asset. Thus, if the option ispath-dependant, each path is followed and only the paths that satisfythe conditions of the option are taken into account. The end results ofeach such path are summarized and lead to the theoretical price of thederivative.

The original Black Scholes model is designed for calculating theoreticalprices for Vanilla options. However, it should be understood that anyreference in this application to the Black-Scholes model refers to useof any model known in the art for calculating theoretical prices ofoptions, e.g., a Brownian motion model, as applied to any type ofoption, including exotic options. Furthermore, this application isgeneral and independent of the way in which the theoretical value of theoption is obtained. It can be derived analytically, numerically, usingany kind of simulation method or any other technique available.

For example, U.S. Pat. No. 6,061,662 (“the '662 patent”) describes amethod of evaluating the theoretical price of an option using a MonteCarlo method based on historical data. The simulation method of the '662patent uses stochastic historical data with a predetermined distributionfunction in order to evaluate the theoretical price of options. Examplesis the '662 patent are used to illustrate that this method generatesresults which are very similar to those obtained by applying theBlack-Scholes model to Vanilla options. Unfortunately, methods based onhistorical data alone are not relevant for simulating financial markets,even for the purpose of theoretical valuation. For example, one of themost important parameters used for valuation of options is thevolatility of the underlying asset, which is a measure for how the rateof the underlying asset fluctuates. It is well known that the financialmarkets use predicted, or “future”, value for the volatility of theunderlying assets, which often deviates dramatically from the historicaldata. In market terms, future volatility is often referred to as“implied volatility”, and is differentiated from “historicalvolatility”. For example, the implied volatility tends to be much higherthan the historical volatility of the underlying asset before a majorevent, such as risk of war, or during and after a financial crisis.

It is appreciated by persons skilled in the art that the Black-Scholesmodel is a limited approximation that may yield results very far fromreal market prices and, thus, corrections to the Black-Scholes modelmust generally be added by traders. In the foreign exchange (FX) Vanillamarket, for example, the market trades in volatility terms and thetranslation to option price is performed through use of theBlack-Scholes formula. In fact, traders commonly refer to using theBlack-Scholes model as “using the wrong volatility with the wrong modelto get the right price”.

In order to adjust the price, in the Vanilla market, traders usedifferent volatilities for different strikes, i.e., instead of using onevolatility per asset, a trader may use different volatility values for agiven asset depending on the strike price. This adjustment is known asvolatility “smile” adjustment. The origin of the term “smile”, in thiscontext, is in the foreign exchange market, where the volatility of acommodity becomes higher as the commodity's price moves further awayfrom the ATM strike.

The phrase “market price of a derivative” is used herein to distinguishbetween the single value produced by some benchmark models, such as theBlack-Scholes model, and the actual bid and offer prices traded in thereal market. For example, in some options, the market bid side may betwice the Black-Scholes model price and the offer side may be threetimes the Black-Scholes model price.

Many exotic options are characterized by discontinuity of the payoutand, therefore, a discontinuity in some of the risk parameters near thetrigger(s). This discontinuity prevents an oversimplified model such asthe Black-Scholes model from taking into account the difficulty inrisk-managing the option. Furthermore, due to the peculiar profile ofsome exotic options, there may be significant transaction costsassociated with re-hedging some of the risk factors. Existing models,such as the Black-Scholes model, completely ignore such risk factors.

Many factors may be taken into account in calculating option prices andcorrections. (Factor is used herein broadly as any quantifiable orcomputable value relating to the subject option.) Some of the notablefactors are defined as follows:

Volatility (“Vol”) is a measure of the fluctuation of the returnrealized on an asset. An indication of the level of the volatility canbe obtained by the volatility history, i.e., the standard deviation ofthe return of the assets for a certain past period. However, the marketstrade based on a volatility that reflects the market expectations of thestandard deviation in the future. The volatility reflecting marketexpectations is called implied volatility. In order to buy/sellvolatility one commonly trades Vanilla options. For example, in theforeign exchange market, the implied volatilities of ATM Vanilla optionsfor many frequently used option dates and currency pairs are availableto users in real-time, e.g., via screens such as REUTERS, Bloomberg,TELERATE, Cantor Fitzgerald, or directly from FX option brokers.

Volatility smile, as discussed above, relates to the behavior of theimplied volatility with respect to the strike, i.e., the impliedvolatility as a function of the strike, where the implied volatility forthe ATM strike is the given ATM volatility in the market. For example,for currency options, a plot of the implied volatility as a function ofthe strike shows a minimum in the vicinity of the ATM strike that lookslike a smile. For equity options, as another example, the volatilityplot tends to be monotonous.

Vega is the rate of change in the price of an option or other derivativein response to changes in volatility, i.e., the partial derivative ofthe option price with respect to the volatility.

Convexity is the second partial derivative of the price with respect tothe volatility, i.e. the derivative of the Vega with respect to thevolatility, denoted dVega/dVol.

Delta is the rate of change in the price of an option in response tochanges in the price of the underlying asset; in other words, it is apartial derivative of the option price with respect to the spot. Forexample, a 25 delta call option is defined as follows: if against buyingthe option on one unit of the underlying asset, 0.25 unit of theunderlying asset are sold, then for small changes in the underlyingoption, assuming all other factors are unchanged, the total change inthe price of the option and the 0.25 unit of the asset are null.

Intrinsic value (IV) for in-the-money knock-out/knock-in exotic optionswith strike K and trigger (or barrier) B, is defined as IV=|B−K|/B.In-the-money knock-out/knock-in options are also referred to as Reverseknock-out/knock-in options, respectively. For a call option, theintrinsic value is the greater of the excess of the asset price over thestrike price and zero. In other words, the intrinsic value ofin-the-money knock out options is the intrinsic value of a correspondingVanilla at the barrier, and represents the level of payout discontinuityin the vicinity of the trigger.

25Δ Risk Reversal (RR) is the difference between the implied volatilityof a call option and a put option with the same delta (in oppositedirections). Traders in the currency options market generally use 25delta RR, which is the difference between the implied volatility of a 25delta call option and a 25 delta put option. Thus, 25 delta RR iscalculated as follows:

25delta RR=implied Vol(25delta call)−implied Vol(25delta put)

The 25 delta risk reversal is characterized by a slope of Vega withrespect to spot but practically no convexity at the current spot.Therefore it is used to price the slope dVega/dspot.

25Δ Strangle is the average of the implied volatility of the call andthe put, which usually have the same delta. For example:

25delta strangle=0.5(implied Vol(25delta call)+implied Vol(25delta put))

The 25 delta strangle is characterized by practically no slope of Vegawith respect to spot at the current spot, but a lot of convexity.Therefore it is used to price convexity. Since the at-the-money Vol isalways known, it is more common to quote the butterfly in which one buysone unit of the strangle and sells 2 units of the ATM option. Like thestrangle, butterfly is also quoted in volatility. For example:

25delta butterfly=0.5(implied Vol(25delta call)+implied Vol(25deltaput))−ATM Vol

The reason it is more common to quote the butterfly is that butterflyprovides a strategy with almost no Vega but significant convexity. Sincebutterfly and strangle are related through the ATM volatility, which isalways known, they may be used interchangeably. The 25 delta put and the25 delta call can be determined based on the 25 delta RR and the 25delta strangle.

Gearing, also referred to as leverage, is the difference in pricebetween the exotic option with the barrier and a corresponding Vanillaoption having the same strike. It should be noted that a Vanilla optionis always more expensive than a corresponding exotic option.

Bid/offer spread is the difference between the bid price and the offerprice of a financial derivative. In the case of options, the bid/offerspread is expressed either in terms of volatility or in terms of theprice of the option. The bid/offer spread of a given option depends onthe specific parameters of the option. In general, the more difficult itis to manage the risk of an option, the wider is the bid/offer spreadfor that option.

Typically traders try to calculate the price at which they would like tobuy an option (i.e., the bid side) and the price at which they wouldlike to sell the option (i.e., the offer side). Currently, there are nomathematical or computational methods for calculating bid/offer prices,and so traders typically rely on intuition, experiments involvingchanging the factors of an option to see how they affect the marketprice, and past experience, which is considered to be the most importanttool of traders. Factors commonly relied upon by traders includeconvexity and RR which reflect intuition on how an option should bepriced. One dilemma commonly faced by traders is how wide the bid/offerspread should be. Providing too wide a spread reduces the ability tocompete in the options market and is considered unprofessional, yet toonarrow a spread may result in losses to the trader. In determining whatprices to provide, traders need to ensure that the bid/offer spread isappropriate. This is part of the pricing process, i.e., after the traderdecides where to place the bid and offer prices, he/she needs toconsider whether the resultant spread is appropriate. If the spread isnot appropriate, the trader needs to change either or both of the bidand offer prices in order to show the appropriate spread.

SUMMARY OF THE INVENTION

The present invention provides a method and a system for calculatingoption prices (e.g., bid and offer prices) and for providing automatictrading capabilities, e.g., via global computer network. Specifically,the method of the present invention enables automatic calculation of thebid and offer prices of options with accuracy comparable to that of anexperienced trader. Thus the invention also enables traders not only tocorrectly evaluate the price of the option, for example, the mid-marketprice of the option, but also to accurately determine the bid-offerspread of the option. Further, since the computation of the bid andoffer prices in accordance with the invention does not involve amorphousfactors and/or trader intervention, investors may transact on theoptions based on the automatically generated bid and offer prices. Byfeeding the model of the present invention with real time market data,the model generates real time market prices for derivatives and,therefore, the model automates the process of buying/sellingderivatives.

In an embodiment of the present invention, the model is used inconjunction with an online trading system whereby on-line transactionsare executed at the prices provided by the model. Liquidity providers,e.g., market makers and banks, may trade at the model prices instead ofproviding their own prices, i.e., they may sell options at the modelgenerated offer price and buy at the model generated bid price avoidingany need for further calculations. Similarly, price-takers, e.g.,hedgers, asset management groups, may execute deals automaticallywithout prior automation of a bank on each transaction individually.

It is appreciated by persons skilled in the art that different types ofasset markets are generally analogous in that they are controlled byanalogous market conditions, e.g., forward rates, interest rates, stockdividends and costs of carry, and therefore, an option-pricing modelwhich is suitable for one type of asset market is generally alsoadaptable to other types of markets, by appropriately interchanging thequantities used by the model with corresponding quantities of adifferent type of derivative. For example, to change the model fromforeign exchange (FX) options to stock options, one would use thedividend rate of the stock in place of one of the interest rates used inthe case of a pair of currencies.

Such adaptation is also possible in cases where the analogy is notsimple, for example, in weather derivatives. To adapt the model of theinvention to any type of option or option-like derivative, instead ofsimply replacing the quantities described below with correspondingquantities of a new type of derivative being computed, the model may beadapted by appropriately modifying its building blocks, which aredescribed below, to accommodate the new type of derivative, andcomputing the price of the derivative based on the new building blocks.It should be appreciated that different option markets share similarbasic principle. Thus, although the invention is described below in thecontext of the foreign exchange (FX) market, the model of the inventionmay be adapted to other option and option-like markets with appropriatechanges, as will be apparent to those skilled in the art.

An embodiment of the present invention calculates bid and offer pricesof an exotic option based on a corrected theoretical value (CTV) of theoption and a bid/offer spread. The CTV may be computed based on aplurality of building blocks, as described below. For example, the CTVmay be calculated based the theoretical value of the exotic option, aset of corrections, and a set of weights, each of which may be computedbased on selection of the various details of the option including thespot, expiration date, class of the option (knock out, knock in, binary,European digital, etc.), strike (when applicable), barrier(s), forwardrate to delivery, volatility for the expiration date, and interest ratesof currencies. It is noted that a more complex exotic option may requireadditional details to define the option. A weight may be computed foreach correction. Some or all of the weights may be time dependent. Thecorrected TV, also referred to herein as the adjusted mid-market price,may be computed as a function of the TV and the weighted corrections, orusing any other suitable function of a plurality of building blocks thatmay reflect risks associated with the option.

To compute the bid/offer spread, a second set of weights may be computedcorresponding to each correction, resulting in a different function ofthe building blocks, as described below. Some or all of the weights maybe time dependent. The bid/offer spread may then be computed as afunction of some base value and the weighted corrections, using thesecond set of respective weights. For example, the base value may bedetermined as the bid/offer spread of a Vanilla option corresponding tothe subject exotic option. The weights applied to the corrections todetermine the bid/offer spread are generally different from the weightsapplied to the corrections for the TV. In alternative embodiments of theinvention, the bid/offer spread may be computing using any othersuitable function of a plurality of building blocks that may reflectrisks associated with the option.

Finally, in computing the bid and offer prices, the model may includecomputation of volatility smile adjustment, for example, using a look-uptable representing volatility smile adjustment in a predefined range.Such a look-up table may be generated by computing the volatility foreach strike value and for each delta value in a predefined set. Ananalogous system and method may be used to compute the bid and offerprices for Vanilla options in addition to exotic options and othercomplex derivatives.

It should be appreciated that the benefit to financial markets fromhaving an accurate model to price derivatives are enormous. First, theaccurate model of the invention enables less experienced users ofderivatives to price them accurately. Second, by virtue of having anaccurate pricing model, the derivatives market is expected to becomemore liquid. Not being able to determine the correct price ofderivatives creates strong dependency on market makers and causes usersto refrain from using derivatives. Third, currently, many corporationsand funds, for example, cannot establish credit lines with vis-à-viseach other and are required to deal only with banks. By having anaccurate model for market prices, any two parties can deal with eachother on a margin basis, even if they do not have mutual credit lines.

In accordance with an embodiment of the invention there is thus provideda method for providing a bid price and/or an offer price of an option onan underlying asset, the method including receiving first input datacorresponding to a plurality of parameters defining the option,receiving second input data corresponding to a plurality of currentmarket conditions relating to the underlying asset, computing aplurality of building blocks based on the first and second input data,at least one of the building blocks being a function of a factor relatedto a risk associated with the option, computing a bid price and/or anoffer price of the option as a function of at least some of the buildingblocks, and providing an output corresponding to the bid price and/orthe offer price of the option. In some embodiments of the invention,computing the bid price and/or the offer price includes computing acorrected theoretical value (CTV) of the option as a first function ofat least some of the building blocks, computing a bid/offer spread ofthe option as a second function of at least some of the building blocks,and computing the bid price and/or the offer price of the option basedon the corrected TV and the bid/offer spread. The plurality of buildingblocks may include at least one building block selected from the groupincluding convexity, risk reversal (RR), shift, gearing, Vega profile,and intrinsic value.

Further, in accordance with an embodiment of the invention, there isprovided a method for providing a bid price and/or an offer price of anoption on an underlying asset, the method including receiving firstinput data corresponding to a plurality of parameters defining theoption, receiving second input data corresponding to a plurality ofcurrent market conditions relating to the underlying asset, computing acorrected theoretical value (CTV) of the option based on the first andsecond input data, computing a bid/offer spread of the option based onthe first and second input data, computing a bid price and/or an offerprice of the option based on the corrected TV and the bid/offer spread,and providing an output corresponding to the bid price and/or the offerprice of the option.

Additionally, in accordance with an embodiment of the invention there isprovided a system for providing a bid price and/or an offer price of anoption on an underlying asset, the system including a server receivingfirst input data corresponding to a plurality of parameters defining theoption and providing an output corresponding to a bid price and/or anoffer price of the option, the server further receiving second inputdata corresponding to a plurality of current market conditions relatingto the underlying asset, and a processor, associated with the server,which computes a plurality of building blocks based on the first andsecond input data, at least one of the building blocks being a functionof at least one factor related to a risk associated with the option, andwhich further computes the bid price and/or the offer price of theoption as a function of at least some of the building blocks. In someembodiments of the invention, in computing the bid price and/or offerprice of the option, the processor computes a corrected theoreticalvalue (CTV) of the option as a first function of at least some of thebuilding blocks, a bid/offer spread of the option as a second functionof at least some of the building blocks, wherein the processor computesthe bid price and/or offer price of the option based on the corrected TVand the bid/offer spread. The plurality of building blocks may includeat least one building block selected from the group including convexity,risk reversal (RR), shift, gearing, Vega profile, and intrinsic value.

Further, in accordance with an embodiment of the invention, there isprovided a system for providing a bid price and/or an offer price of anoption on an underlying asset, the system including a server receivingfirst input data corresponding to a plurality of parameters defining theoption and providing an output corresponding to a bid price and/or anoffer price of the option, the server further receiving second inputdata corresponding to a plurality of current market conditions relatingto the underlying asset, and a processor, associated with the server,which computes, based on the first and second input data, a correctedtheoretical value (CTV) of the option and a bid/offer spread of theoption, and which further computes, based on the CTV and bid/offerspread, the bid price and/or the offer price of the option.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be understood and appreciated more fully fromthe following detailed description of a preferred embodiment of theinvention, taken in conjunction with the accompanying drawings of which:

FIG. 1 is a flow chart illustrating an overview of a method for pricingoptions in accordance with an embodiment of the present invention;

FIGS. 2A-2D are sequential flow charts schematically illustrating analgorithm for calculating bid/offer prices of foreign exchange (FX)options in accordance with an embodiment of the present invention; and

FIG. 3 is a schematic block diagram illustrating a system for pricingoptions in accordance with an embodiment of the present invention.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT

A preferred embodiment of the present invention is described in thecontext of a model for calculating the market value (market price) of aforeign exchange (FX) exotic option. It should be appreciated, however,that models in accordance with the invention may be applied to otherfinancial markets, and the invention is not limited to foreign exchangeoptions or exotic options. One skilled in the art may apply the presentinvention to other options, e.g., stock options, or other option-likefinancial instruments, e.g., options on futures, or commodities, ornon-asset instruments, such as options on weather, etc., with variationas may be necessary to adapt for factors unique to a given financialinstrument.

In the embodiment described herein below, bid/offer prices are computedfrom a corrected theoretical value (TV) of an option and the bid/offerspread for that option. Computations for the corrected TV and bid/offerspread apply derivatives (partial derivatives up to second order) tofactors readily available in the market. The factors include, forexample, gearing (where the trigger is cancelled by setting it to zero,when the trigger is below the asset rate, or to infinity, when thetrigger is above the asset rate) and the change in the profile of theVega. Instead of trying to assess probabilities, the model presentedherein enables assessment of the risk management cost of the option andof the compensation required by a trader in trading the option. Incontrast to the Black-Scholes model, which is a probabilistic model, theapproach of the present invention is based on determining whatcorrections must be added to the theoretical value of an option in orderto compensate for the risk of the option in the trading book, e.g., theoption portfolio run by the market maker. The key factors selected toachieve the goals of the models are referred to as building blocks.

The model of the present invention takes into account many factors thatthe Black-Scholes model ignores, e.g., factors that are related totransaction cost of re-hedging. For example, in the model describedherein, the re-hedging cost of the Vega of the exotic option may beexpressed in terms of the convexity cost of the option. By having a longconvexity, i.e., a positive convexity in the trader book, a trader canearn money by selling volatility (e.g., selling ATM options) when thevolatility is higher and buying volatility when it is lower, withouttaking a position. The shift of the trigger represents the probabilityof an option being near the trigger at a time close to the maturity ofthe option, at which point the re-hedging cost is the most expensiveand, thus, the option is most risky.

Since the trader is typically delta hedged, at a knock-out event, theseller of a knock-out option should remove the delta hedge in astop-loss trade, e.g., by buying back the underlying asset when themarket rises or selling the underlying asset when the market declines.In-the-money knock out options are characterized by a growing deltadiscontinuity towards expiration of the option. As the time of maturityapproaches, the delta re-hedging cost near the trigger may risedrastically, and the price of the option reflects the risk near thetrigger. It should be noted that the shift of the trigger correction isalways positive and thus needs to be properly gauged to express therisk. The gearing reflects some aspects of the time decay of the exoticoption because the price of the option will converge to the price of thecorresponding Vanilla option if it is not knocked out. Typically, theshorter the option, the more re-hedging is required to account for timedecay.

In accordance with an embodiment of the present invention, twoquantities are calculated separately, namely, the adjusted mid-marketprice and the bid/offer spread. According to this embodiment, separatecalculations are used for computing the two quantities. The adjustedmid-market price is defined as the middle (i.e. the average) between thebid price and the offer price. As discussed above, the Black-Scholesmodel provides one price that may be referred to as theoreticalmid-market price or theoretical value (TV). The adjusted mid-marketprice provided by the present invention may be regarded as an adjustmentto the Black-Scholes price. Thus the adjusted mid-market price of thepresent invention may also be referred to as the corrected theoreticalvalue (CTV). It should be appreciated that, since the final outputs ofthe model, typically provided to the user, are the bid and offer prices,as discussed below, the use of mid-market price as a reference point forthe model of the invention merely for convenience and may be replaced byother arbitrary reference points, for example, a higher or lower valuecorresponding to some known function of to the mid-market price. The useof mid-market price as a reference for the computations is preferredsimply because existing theoretical models for calculating prices ofoptions, such as the Black Scholes model, are typically intended forcalculating theoretical mid-market values.

The bid/offer spread, computed according to the preferred embodiment,reflects the risk that is related to re-hedging transaction costsentailed in the option. The building blocks used for corrections in thecalculation of the bid/offer spread may be similar to those used tocalculate the mid-market price because both the mid-market price and thebid/offer spread are related to the risk of the option. However, thebuilding blocks are used with different relative weights in the twocalculations. For example, in some cases terms may “cancel out”calculating the adjusted mid-market price, but, the same terms may havea cumulative effect in calculating the bid/offer spread as separateindependent hedging costs that increase the spread.

By way of an overview of the preferred embodiment, referring to FIG. 1,at stage 110 the model calculates a theoretical value (TV) using acombination of known algorithms, e.g., based on the Black-Scholes model,or any model assuming that spot undergoes a Brownian motion pattern.This initial TV may be computed in an analytical method or usingnumerical calculations, as are known in the art. The Black-Scholes modelis used in an exemplary embodiment because it is a common benchmark inthe industry for pricing derivatives in cases where the underlying assetis assumed to follow a Brownian motion (a stochastic process). Theinputs for the TV may include expiration date, class of the option,e.g., knock out, knock in, binary, European digital, etc., strike (whenapplicable), barrier(s), spot, forward rate to delivery, volatility forthe expiration date, and interest rates of currencies. At stage 112, themodel calculates corrections and weights to apply to the TV to generatethe adjusted mid-market price, also referred to herein as corrected TV(CTV).

In this exemplary embodiment, the building blocks include convexity,risk reversal, intrinsic value, gearing, shift, and Vega. Thecorresponding corrections may then include convexity correction, riskreversal correction, intrinsic value correction, gearing correction,shift correction, and Vega profile correction, as described below.Weights are calculated for each correction where some of the weights maybe time dependent. At stage 114, the corrections and correspondingweights are applied to the TV to generate the corrected TV. At stage116, the model calculates the bid/offer spread by combining thedifferent building blocks of the bid/offer spread, e.g., some basevalue, the Vanilla bid/offer spread, and the various calculatedcorrections and weights, some of which weights may be time dependent,and which may be different from the weights used to compute thecorrected TV. Finally, at stage 118, the bid and offer prices arecomputed from the corrected TV and the bid/offer spread provided by themodel. The preferred embodiment is demonstrated as applied toin-the-money barrier options, by way of example. It should beappreciated that, with appropriate changes, the invention may be appliedto any other type of option or option-like financial derivative known inthe art.

Reference is made to FIGS. 2A-2D, taken sequentially, whichschematically illustrate a method for computing adjusted mid-marketprice and bid/offer spread, and bid and offer prices of a foreignexchange (FX) option, in accordance with an embodiment of the invention.As shown in FIG. 2A, the inputs for the calculation indicated at block12 preferably include many significant details of the subject option orthe relevant market. Option details include information derived from therelevant market, referred to herein as market conditions, as well asdetails defining the option, referred to herein as parameters, which maybe specified by the user. Market conditions include market informationthat describes or relates to the subject option, as well as marketinformation not specific to the subject option. Examples of marketconditions include spot, volatility, forward rate, and interest rates.Parameters include, for example, strike, trigger value(s), andexpiration date(s). The parameters may also include an identification ofthe type of option, an identification of the underlying asset, e.g., thecurrencies being exchanged, and other information defining the option.For example, to compute the price of a window knock-out option, theoption details may also include the date on which the trigger isactivated and the date on which the trigger is deactivated. Values formarket conditions, e.g., the current interest rates, forward rates, andthe ATM volatility, may be obtained from information available in themarket, as is known in the art. The market information is based onassets that are continuously traded in the market and their prices areavailable in different forms. For example, the inputs may be based oninformation taken from screens of market data provided by companies suchas REUTERS, Bloomberg, Telerate, etc., and/or directly from brokers,e.g., over the telephone.

Block 14 indicates the computation of the theoretical value (TV) of theoption being priced. The algorithm for computing the TV may be based onBlack-Scholes or similar models consisting of analytic formulas orsimulation methods as are known in the art. In some cases, for example,When computing double knock-out options, the computation may involvesumming over infinite series; however, due to the fast convergence ofsuch infinite series, it is generally sufficient to sum the first tenelements of such series. For example, a double knock-out option, whichis similar to knock-out option but has two barriers (one above thecurrent spot level and one below), involves summing over an infiniteseries, but yields acceptable results when only the first ten elementsof the series or less are summed.

Continuing reference to FIG. 2A, block 16 indicates the calculation ofthe strikes and volatility (denoted “Vol”) of 25 delta call and put,respectively, i.e. the strikes for which the delta for the givenvolatility is 25 percent. The implied volatility of the 25 delta calland put may be derived directly from the 25 delta RR and 25 deltabutterfly (strangle). Block 18 indicated the input of these two valuesthat may be obtained from the market conditions. As mentioned in thebackground section above, the 25 delta RR and 25 delta strangle(butterfly) are commodities in the options market and quotes for thoseinputs are readily available from well known online sources, as is thecase for the ATM volatility.

Block 20 indicates the calculation of derivatives of Vega including theconvexity of the 25 delta strangle, the slope of Vega over spot of the25 delta risk reversal, as well as the price per convexity and price perrisk reversal. These quantities may be used to gauge the correspondingderivatives of the exotic option. By comparing the premium (i.e., theprice) of the 25 delta strangle to the premium of the 25 delta stranglewith ATM volatility, the model may compute the price of a unit ofconvexity, denoted “Price(convexity)”. By comparing the price being paidfor 25 delta RR versus ATM, the model may calculate the price of oneunit of dVega/dSpot, also denoted “Price(RR)”. At this stage of thecomputation, all the relevant values for 25 delta are computed. Next,the strikes and volatility values for other values of delta, within apreset range, may be computed. Block 22 indicates the calculation ofVega from the TV used in the computation indicated by block 20.

As indicated at block 24, the strikes and volatility for each delta arecomputed and may be calculated directly or arranged in memory, e.g., inthe form of a look-up table, for reference later in the computation. Thetable generated indicates the relationship between the strike andvolatility so that the volatility for a given strike may be looked up inthe table. The algorithm for generating this look-up table, alsoreferred to as volatility smile adjustment, is described in detailbelow, following the description of computing the bid and offer prices.The novel volatility smile adjustment mechanism in accordance with thepresent invention is not limited to computation of exotic options, asdescribed herein; rather, this novel mechanism has a generalapplicability and may also be used independently for obtaining themid-market price of Vanilla options.

Referring now to FIG. 2B, the corrections and weights are calculated forthe particular option, as indicated at block 32. The corrections aredenoted, respectively, as follows: convexity correction at block 38;risk reversal correction at block 44; intrinsic value correction atblock 40; gearing correction at block 34; shift correction at block 42;and Vega profile correction at block 36. It should be noted by oneskilled in the art that additional corrections may be used in thecomputation of the corrected TV and bid/offer spread, and similarly notall of the corrections listed herein need be used to produce valuableresults for a particular option. The corrections used in the exemplaryembodiment described herein may be defined as follows:

-   -   (a) Convexity correction=convexity*Price(Convexity)    -   (b) Risk reversal correction=(dVega/dSpot)*Price(RR)    -   (c) Intrinsic value correction=intrinsic value    -   (d) Gearing correction=(TV(Vanilla)−TV(exotic))*Ratio    -   wherein Ratio is a function dependent on the ratio between        TV(Vanilla) and TV(exotic), for example, as follows:    -   Ratio=sqrt [(TV(Vanilla)/TV(exotic))/8.5] if        V(Vanilla)/TV(exotic)<=8.5 and        Ratio=exp(−(TV(Vanilla)/TV(exotic)−8.5)/80) otherwise.    -   (e) Shift correction is a function of the change in the TV when        the barrier is shifted and the change in TV when the expiration        is shifted. The following formula may be used:    -   Shift=abs(TV(exotic with barrier B and expiration t)−TV(exotic        with barrier B′ and expiration t′))    -   wherein |B′−K|/B′=1.05*|B−K|/B and t′=t+1 day.    -   (f) Vega Profile correction is a function of the behavior of the        Vega of the exotic option as a function of the spot, e.g., in        several spot points. The Vega Profile correction quantifies the        shape of the Vega as a function of the spot. The Vega profile        corrections may be calculated in 3 steps, providing different        aspects of the shape of the Vega profile, namely, Profile1,        Profile2, and Profile3.        -   1. Using K as the strike of the exotic option and B as the            trigger (barrier) of the exotic option:            -   Profile1=Smile(K)+Smile(B)*(Vega(exotic)−Vega(Vanilla                with strike K))/Vega(Vanilla with strike B)        -   2. Profile2 is determined by replicating the Vega of the            exotic option with, for example, three Vanilla options with            strikes K, Kmin, and B, at spot points, S, Smin and B,            respectively. The replication is performed by looking for            numbers p, q, and r, for which the following equation is            satisfied (at the spot points, S, Smin and the barrier, B):            -   Vega(exotic at spot X)=p*Vega(Vanilla with strike K, at                spot X)+q*Vega(Vanilla with strike Kmin, at spot                X)+r*Vega(Vanilla with strike B, at spot X)            -   wherein X=S, Smin, and B, sequentially.            -   It is noted that (Vega(exotic at spot B) is equal to                zero because the option terminates at spot B.            -   In the above equations, Vega(Vanilla with strike K, at                spot X) refers to the calculation of the Vega of the                vanilla option at spot X using the volatility which                corresponds to the strike through the Volatility Smile,                i.e., find Smile(K), then find the volatility VolK such                thatTV(vanilla with strike K, volatility                VolK)=TV(Vanilla with strike K and ATM                volatility)+Smile(K), then, using VolK, find Vega                (Vanilla with strike K and volatility VolK)            -   The numbers p, q, and r, are obtained by solving the                three equations with three unknowns above. Accordingly:            -   Profile2=p*Smile(K)+q*Smile(Kmin)+r*Smile(B)        -   3. Profile3=Smile(Kmin)*Ratio(Vega (Smin)/Vega (Kmin))    -   After calculated Profile1, Profile2 and Profile3, the following        formula may be applied:    -   TotalProfile=minimum        ((1−Ptouch)*(0.115*Profile1+0.55*Profile2),0) if Profile3<0, and        TotalProfile=0 otherwise.    -   wherein Ptouch(t) is the probability of touching the trigger        prior to time t.    -   The Vega Profile correction may then be obtained as follows:    -   Vega Profile correction=Maximum(TotalProfile,        Profile3)+(1−exp(3Π/2*t)*Minimum(TotalProfile, Profile3) if        Profile3<0, and    -   Vega Profile Correction=0 otherwise.

In this embodiment, there is a building block corresponding to eachcorrection indicated by blocks 34-44. The building blocks and othervalues needed to compute the corrections are based on the valuesdetermined at blocks 16-24 of FIG. 2A, using option parameters andmarket conditions. Convexity is defined as dVega/dVol. Price(Convexity)is the average Vega of the 25 delta call and put Vanilla optionsmultiplied by the butterfly and divided by the dVega/dVol of the 25delta strangle. Risk Reversal is defined as dVega/dSpot. Price(RR) isthe average Vega of the 25 delta call and put Vanilla options multipliedby the RR and divided by dVega/dSpot of the 25 delta Risk Reversal.Intrinsic value is the distance between a given trigger value and thestrike, normalized by the trigger value. Gearing is the difference inprice between the exotic option with the given trigger and acorresponding Vanilla option with the same strike. TV(exotic) is thetheoretical value of the original exotic option as calculated by theBlack-Scholes model. TV(Vanilla) is the theoretical value of thecorresponding Vanilla option, i.e. the option with the same parametersexcept for the triggers. Ratio(TV(exotic)/TV(Vanilla)) is the ratiobetween TV(exotic) and TV(Vanilla) subject to a cut-off when the ratioexceeds a predetermined value, for example, a value between 6 and 12.

The gearing correction is proportional to the difference and ratiobetween the theoretical values of the exotic option and a Vanilla optionwith the same parameters. For each exotic option with a strike, there isa corresponding Vanilla option. For example, a knock-out option has atrigger or barrier. A Vanilla option corresponding to this exotic optionwill have the same maturity time, spot, strike, etc. but no barrier.Since adding a knock out barrier limits the validity of the option,e.g., it is possible that the exotic option knocks out (i.e.,terminates) while the corresponding Vanilla option ends up in the money,the exotic option would generally be less expensive than thecorresponding Vanilla option. The gearing correction depends on theratio and difference between the TV of the Vanilla option and the TV ofthe exotic option.

The shift correction is a function of two values: the change in TV whenthe trigger is shifted, and the change in TV when the expiration isshifted. The shift correction function may be, for example, the maximumof these two values. Alternatively the function may be the sum of thesetwo values. The first value may be computed by shifting the trigger sothat the intrinsic value is increased by a certain percentage, e.g., 5percent, and determining the resultant change in the TV. The secondvalue may be computed by shifting the expiration by, for example, oneday, and determining the change in TV. The shift correction is a measureof the sensitivity of the theoretical value (TV) of the option price tochanges in the trigger value and expiration.

The Vega profile correction requires characterizing the profile of theVega with respect to the spot. Such characterization may involve, forexample, the following factors: Vega (Smin); Vega (Kmin); andSmile(Kmin). Vega (Smin) is the Vega of the barrier option at a spot,Smin, which produces the minimum value of Vega. In other words, Smin isthe minimum of the Vega of the exotic option with respect to the spot.Vega (Kmin) is the Vega of the Vanilla option with strike, Kmin.Smile(Kmin) is the smile adjustment, i.e., the adjustment of the priceof a Vanilla option, with a strike Kmin. Kmin may be computed using thefollowing equation:

Smin=Kmin*(current Forward rate)/(current spot rate)

Thus, Kmin is the strike that yields a forward rate of Smin at currentinterest rates.

The volatility of the option may be determined by finding, in thelook-up table, denoted by block 30, the volatility for the computedstrike, i.e. Kmin. It should be appreciated by persons skilled in theart that characterization of the Vega profile with respect to the spotmay also be performed using other suitable parameters, for example, incertain cases, instead of using one strike value (e.g., Kmin), asdescribed above, more than one strike value may be used to approximatethe Vega profile.

Once the above-described corrections are computed, they are added to theTV, either directly or with some restrictions, using time dependentweights as described in detail below, producing the total corrected TV.The weights of the corrections generally reflect the risk involved ineach correction. For example, some of the corrections, e.g., the gearingcorrection, have an enhanced influence close to the maturity of theoption, but very small influence when the option is far from maturity.Other corrections, e.g., the convexity correction, have less of aninfluence close to the maturity. Therefore the weights are generallytailored to adjust for the specific risk versus time-to-maturitybehavior of each of the corrections.

The motivation for adding weighted corrections to the TV, in accordancewith the invention, is partly based on the realization by the inventorsthat models such as the Black-Scholes model underestimate theprobability of reaching a far spot level when time to maturity is long.In reality, the probability for a far knock out is generally higher thanthat anticipated by the Black-Scholes formula. This is part of thereason for the decay of most of the factors with time to maturity beyonda certain level. This type of adjustment may be particularly valuablewhen calculating prices of “one touch” options, i.e., options where thebuyer receives a certain payout if the spot touches the barrier,

For the class of in-the-money knock out options, (also called ReverseKnock out options) the weights used for the computation of the correctedTV, wherein Ca denotes the time-dependent weight for correction (a), Cbdenotes the time-dependent weight for correction (b), etc., are asfollows:

-   -   Ca=0.61*exp(−0.4*t)*(1−Ptouch(t/L)*W*LT    -   wherein L=2 if t>1, L=1 if t<1/12, and L=1+(t−1/12)*12/11        otherwise;    -   W=(0.5+0.5*(25Delta butterfly−0.5))/25Delta butterfly if 25Delta        butterfly>=0.5, W=1 otherwise; and    -   LT=2*sqrt(t) if t<0.25, and LT=1 otherwise.    -   Cb=0.6*Π*sqrt(t)*exp(−t*Π/2))*(1−Ptouch(t/L))    -   wherein L is as defined above.    -   Cc=0    -   Cd=0.045*minimum(1, 4.5*exp(−12t)+exp(−1))    -   Ce=0.135*t+0.1125 if t<1, and 0.2475 otherwise    -   Cf=(0.5+exp(−2*Π*0)*(1−exp(−2*Π*t))        wherein Ptouch(t) is the probability of touching the trigger        prior to time t, denoted in years (E.g. for one year t=1.)

As indicated at blocks 46 and 50, following the computation of thecorrections and weights for a given option, the corrected TV (CTV),i.e., the adjusted mid-market price, may be computed using all theblocks. In combining the building blocks of the invention, severalissues should be addressed. First, the Risk Reversal correction and theConvexity correction are local in the sense that they relate to the nearvicinity of the spot range, whereas the Vega Profile correction is aglobal correction, i.e., the correction takes into account the Vega ofthe exotic option in a relatively wide spot area. In a certain area ofthe spot range, part of the quantified value of the profile is alreadytaken into account in the Risk Reversal correction and the convexitycorrection and vice versa and, therefore, duplications should beavoided. For example, if the value of the profile is determined toresemble a certain Vanilla option, then the Vega Profile correction maytake into account the Smile adjustment of that Vanilla and, therefore,adding the Risk Reversal correction and Convexity correction couldresult in double counting, in view of the smile adjustment mechanismused. Second, the Risk Reversal correction and the Convexity correctionare linear in the 25 Delta RR and butterfly. In addition to removing thelinearity in these factors, the Vega Profile correction should includehigher order derivatives and should remove the linearity, as long as itis combined properly with the Risk Reversal Correction and Convexitycorrection. Third, in some cases, the Gearing correction and the Shiftcorrection may overlap the quantified risk involved in ending up nearthe trigger. This may happen particularly in the vicinity of the spot,where both these corrections tend to maximize.

Taking the above considerations into account, the following mechanismmay be used to combine the building blocks, wherein building block areadded one by one. For this process, the following corrections aredefined:

-   -   a—Convexity correction    -   b—Risk Reversal correction    -   c—Intrinsic value correction    -   d—Gearing correction    -   e—Shift correction    -   f—Vega Profile correction        1. Combining the Risk Reversal Correction with the Vega Profile        Correction:        If Risk Reversal correction>=0 and Vega profile correction<=0        then Correction1=Cb*b+Cf*f

Otherwise

Correction1=exp(−3/2**LLT*maximum(Cb*b, Cf*f)+minimum(Cb*b, Cf*f)wherein LLT=4*t if t<0.25, and LLT=1 otherwise.2. Combining Correction1 with the Convexity Correction:If Convexity correction<0 and Correction1<0, then:set LowCutOff(t)=8% if t<=1/12, LowCutOff=15% if t>1, and

-   -   LowCutOff=8%+7%*(t−1/12)*12/11 otherwise.        Set HighCutOff(t)=18% if t<=1/12, HighCutOff=20% if t>1, and    -   HighCutOff=18%+2%*(t−1/12)*12/11 otherwise.

Set ConvexityRatio=Ca*a/TV(exotic).

If ConvexityRatio>LowCutOff(t), thenset ProfileFactor=maximum(0.5,1−0.5*(ConvexityRatio−LowCutOff(t))/(HighCutOff(t)−LowCutOff(t))).If Convexity correction<0 and Correction1>0, then ProfileFactor=1.If Convexity correction>0 and Correction1<0, then ProfileFactor=1.Finally, Correction2=ProfileFactor*Correction1+a*Ca4. Combining Correction2 with the Shift correction and the Gearingcorrection:

Set Fshift=0 if Ce*e=<0.07%, Fshift=1 if Ce*e>=0.09%, and

-   -   Fshift=(Ce*e−0.07%)/0.02% otherwise.

Set Fgearing=0 if Cd*d=<0.07%, Fgearing=1 if Cd*d>=0.09%, and

-   -   Fgearing=(Cd*d−0.07%)/0.02% otherwise.        Fcombine(t)=0 if t=<0.019, Fcombine(t)=(t−0.019)/(0.0641) if        0.019=<t<=1/12,        Fcombine(t)=1 if 1/12=<t=<0.41, Fcombine(t)=1−(t−0.41)/(0.59) if        0.41=<t<=1, and Fcombine(t)=0 otherwise.        Correction3=Correction2+Ce*e+Cd*d−Ptouch*(Ce*e+Cd*d−(0.16%+0.05%*minimum        (t,1))*F shift*Fgearing*Fcombine(t)    -   Finally, Total Correction=Correction3, and:        -   CTV=TV+Total Correction            wherein TV is the theoretical value.

Referring now to FIG. 2C, the bid/offer spread may be computed based onthe same set of corrections along with a different set of weights, someof which may be time dependent. These new weights may be functions ofthe underlying corrections. Using the newly calculated weights, thecorrections are summed, yielding the bid-offer spread. Block 52indicates computing the new weights, blocks 54-64 indicate thecorrections for computing the sum, block 66 indicates summing of theweighted corrections, and block 68 indicated computing the bid/offerspread.

The bid/offer spread of the exotic option may depend also on thebid/offer spread of the corresponding Vanilla option in the currentmarket conditions. The spread of an exotic option per Vega is generallywider than that of the corresponding Vanilla, for example, by a factorof about 1.5*(Vega of the exotic)/(Vega of the ATM Vanilla), or higher.Therefore, the bid/offer spread of the ATM Vanilla option may be used asa base value in the bid/offer spread computation of the preferredembodiment. It should be appreciated, however, that other suitablefactors may be used in addition to or instead of the Vanilla bid/offerspread in formulating the base value for the bid/offer spreadcalculation in accordance with the invention. The remaining factors maybe the same as those used for computing the corrected TV, as describedbelow, or different building blocks may be used for the bid/offer spreadcomputation, based on the principles discussed above, to adapt forparticular option types. The corrections, as applied to the computationof the bid/offer spread, are indicated in FIG. 2C as follows: convexitycorrection at block 58; risk reversal correction at block 64; intrinsicvalue correction at block 60; gearing correction at block 54; shiftcorrection at block 62; and Vega Profile correction (also referred to as“Vega correction” for short) at block 56.

Since the bid/offer spread is related to the risk/transaction cost ofre-hedging, the corrections have similar properties as those used forthe adjusted mid-market price. However the different corrections areadded in the bid/offer spread calculation with absolute values becausethe transaction costs involved in re-hedging the different parametersare generally independent. For example, an option may have a positiveconvexity, which lowers the price, and a negative risk reversal, whichraises the price, causing an over-all small change in the CTV. However,hedging the convexity is independent of hedging the risk reversal and,therefore, these two corrections result in a wider bid/offer spread. Inthis regard, the “double counting” considerations discussed above shouldbe taken into account in calculating the Total Correction. The weightsapplied to the corrections are denoted Sa, Sb, Sc, Sd, Se, and Sf,respectively. These weights are calculated as follows:

Sa=1/1.55

Sb=0.2

Sc=minimum(1,(1−TV(exotic)/TV(Vanilla))/0.15)

Sd=0.018*exp(−t)

Se=0.45*exp(−1.6t)

Sf=0.2

In an embodiment of the invention, in order to calculate the Bid/Offerspread, the building blocks may be combined in three steps, as follows.1. The convexity correction is added to the combined correction of theRisk Reversal correction and Vega Profile correction, which combinedcorrection is computed as discussed above with reference to TotalCorrection.Thus, Spread1=Sf*abs(Correction1)+Sa*abs(a)+minimum (Sc*c, 0.1%)Wherein “abs” denotes absolute value. It should be noted that if theRisk Reversal correction and the Vega Profile correction have oppositesigns, then Sf*abs(Correction1) becomes abs(Sf*Cf*f+Sb*Cb*b).2. To combine the shift correction and the gearing correction, thefollowing parameters are defined:ShiftTrim=e if e=<0.15% and ShiftTrim=0.15%+0.5*(e−0.15%) otherwise.GearingTrim=Sd*d if Sd*d=<0.08%, and GearingTrim=0.08%+0.5*(Sd*d−0.08%)otherwise.

Spread2=Spread1+Se*((1−Fcombine(t))*e+Fcombine(t)*ShiftTrim)+Srd*((1−Fcombine(t))*d+Fcombine(t)*GearingTrim/Sd)

3. Add standard Vanilla Bid/offer spread.Finally we obtain:Bid/offer spread=(0.7+0.42*exp(−1.1t))*Spread2+VanillaSpread (K)+maximum(VegaATM*ATM Volatility Bid/Offer spread−VanillaSpread(K), 0)*minimum(1, Vega of the exotic option/VegaATM)wherein VanillaSpread (K) is the bid/offer spread of the Vanilla optionwith the same strike as the exotic option and VegaATM is the Vega of theATM Vanilla option.

Reference is now made to FIG. 2D. After computing the bid/offer spread,the bid and the offer prices are computed, as indicated at block 70, bysubtracting and adding, respectively, one half (0.5) of the bid/offerspread to the average price calculated. Hence, as denoted at block 72,the bid is the adjusted mid-market price (CTV) minus half the spread,and the offer is the adjusted mid-market price (CTV) plus half thespread, as indicated at block 74.

As discussed above with reference to calculating the smile adjustmentfor Vanilla options and the generation of the look-up table at block 24(FIG. 2A), the algorithm for computation of the volatility smileadjustment for Vanilla options has general applicability, as isdemonstrated below in the context of calculating the bid/offer spreadfor the Vanilla option. Therefore, the present invention also includes amethod for pricing Vanilla options for any given strike. The volatilitysmile adjustment is calculated from the volatility of an At-The-Money(ATM) Vanilla option and 25Delta call and put options. The factors usedin calculating the volatility smile include Vega, dVega/dSpot (i.e.,risk reversal), and dVega/dVol (i.e., convexity). Vega is the partialderivative of the option value (price) with respect to the volatility.dVega/dSpot is the partial derivative of the Vega with respect to thespot, and dVega/dVol is the partial derivative of the Vega with respectto the volatility.

Two additional factors, derived from market conditions, used incalculating the volatility smile adjustment include 25delta butterflyand 25delta Risk Reversal, both measured in units of volatility. Itshould be noted that for the purpose of the algorithm for the VolatilitySmile, in accordance with the invention, it is not important that theinputs be 25delta butterfly and 25delta Risk Reversal. The input mayalso be of any two other strikes for which market volatility may beobtained, or even two pairs of strikes, for which the total premium inthe market is known. Since the model of the invention applies aniteration method, the 25delta butterfly and 25delta Risk Reversal can bededuced from the data. For example, in some markets, e.g. currencyoptions, the 25delta butterfly is traded in the market with strikes thatcorrespond to the same volatility for both the call and the put options.In such case, the model can obtain the “true” 25delta butterfly byiteration, so that the total premium of the two strikes coincides withtheir smile. As another example, the ATM volatility and the 25 delta RRand butterfly for equity derivatives may be inferred from the price ofthree options that are traded in the exchange for the same expirationdate. This versatility exemplifies the general applicability of themodel of the present invention.

As discussed above, the 25delta butterfly and 25delta Risk Reversalfactors are defined or calculated as follows:

25delta butterfly=0.5*(implied Vol(25delta call)+implied Vol(25deltaput))−ATM Vol.

25delta risk reversal=implied Vol(25delta call)−implied Vol(25delta put)

The volatilities of the 25delta options may be calculated from these twofactors. Using multiple iterations, the entire volatility smile may becalculated, e.g., a look-up-table may be constructed linking strikes tocorresponding deltas and volatility values. Hence, starting with thevolatility of an at-the-money option, which is known, volatilities forthe option at various different deltas (i.e., not only ATM) may becomputed using the 25 delta butterfly and 25 delta risk reversal. Eachset of strike-volatility-delta is unique and may be included in thelook-up-table for easy reference later in the algorithm.

Thus, the smile adjustment for Vanilla options may be computed startingwith the following inputs: 25 delta risk reversal; 25 delta strangles(butterfly); ATM volatility; spot; forward rate; and interest rates. Thealgorithm for calculating the smile adjustment may include the followingsteps:

-   -   1. For a given delta, D1, find the strikes of the D1 delta        strangle. If D1 is less than a predetermined value, e.g., 30,        use the 25 delta implied volatility, obtained from the RR and        the strangles; otherwise use the ATM implied volatility.    -   2. Calculate the following: dVega/dVol of the D1        strangle*Price(convexity) and dVega/dSpot of D1 (RR)*Price(RR);        wherein Price(convexity) and Price(RR) are calculated from the        25 delta strangle and 25 delta RR, as discussed above.    -   3. Calculate a desired premium for D1 strangle over its premium        with ATM volatility, by requiring the same price for one unit of        convexity as the price for one unit of 25 delta strangle. Repeat        this calculation for the dVega/dSpot.    -   4. Adjust the implied volatility of the D1 strikes to fulfill        the same price for a unit of convexity as for the 25 delta        butterfly, and the same for a unit of dVega/dSpot as for the 25        delta risk reversal.    -   5. Calculate new strikes corresponding to delta D1 with the        volatility in step 4.    -   6. Repeat steps 3-5 sequentially until convergence is achieved.    -   7. Set the last volatility obtained as the implied volatility        for D1 strikes.    -   8. Repeat steps 1-7 for other deltas to create a look-up-table        of strikes and their implied volatility.    -   9. For strikes positioned between those in the look-up-table,        use interpolation based on the values of the look-up-table. It        should be noted that the smile adjustment for a given strike is        independent of whether the option is a call option or a put        option.

In an alternative embodiment, the method of the present invention can beused to calculate SmileAdjustment for stike K directly, i.e., not basedon a look-up-table, as follows:

-   -   1. For a given strike K, calculate delta D1. Find the strike        K_(—)1 so that K and K_(—)1 are the strikes of the D1 delta        strangle. If D1 is below a predetermined value, e.g., 30, use        the 25 delta implied volatility obtained from the RR and the        strangles, i.e. the 25 delta call volatility for        maximum(K,K_(—)1) and the 25 delta put volatility for        minimum(K,K_(—)1). Otherwise, use the ATM implied volatility for        both strikes.    -   2. Calculate the following: dVega/dVol of the D1        strangle*Price(convexity) and dVega/dSpot of D1 (RR)*Price(RR),        wherein Price(convexity) and Price(RR) are calculated from the        25 delta strangle and 25 delta RR, as discussed above.    -   3. Calculate a desired premium for D1 strangle over its premium        with ATM volatility, by requiring the same price for one unit of        convexity as the price for one unit of 25 delta strangle. Repeat        this calculation for the dVega/dSpot.    -   4. Adjust the implied volatility of the D1 strikes to fulfill        the same price for a unit of convexity as for the 25delta        butterfly, and the same for a unit of dVega/dSpot as for the 25        delta risk reversal.    -   5. Calculate new Delta D2 of strike K with the volatility        obtained for stike K in step 4. Calculate the strike K_(—)2, so        that K and K_(—)2 are the strikes of the D2 delta strangle with        the volatility for K_(—)1 in step 4.    -   6. Repeat steps 3-5 sequentially until the volatility of strike        K converges.    -   7. Set the last volatility obtained as the implied volatility        for strike K.

The computation presented above for the bid/offer spread has generalapplicability as is demonstrated by the following algorithm forcomputing the bid/offer spread for Vanilla options. The input for thiscomputation may include the bid/offer spread of the ATM volatility, asis known in the art. The market data input may include both bid andoffer ATM volatilities. The algorithm for computing the bid/offer spreadmay be as follows:

-   -   1. Calculate the bid/offer spread of the ATM option in basis        points (“bp”), i.e., in units corresponding to 1/100 of a        percent of the quantity being traded. This may also be        approximated using the following formula:

SpreadATM=Vega(ATM)*(bid/offer spread of volatility).

-   -   2. Calculate the smile adjusted volatility for a given        strike, K. Calculate the delta that corresponds to strike K,        denoted delta (K), by taking the call if the strike is above the        ATM Strike (roughly equal to the forward rate) and take the put        if the strike is below the ATM Strike. As a reminder, the ATM        strike is the strike for which the sum of the delta of the call        option and the delta of the put option is zero.    -   3. Calculate the bid offer spread for strike K in basis points        (bp), as follows:        Spread(K)=VegaATM*ATM Volatility Bid/Offer spread*G(Delta, TV)        wherein G(Delta, TV)=1 if Delta(K)>=7%,        G(Delta, TV)=1−0.645*exp(−15*(VanillaTV(K)+maximum(Smile(K),0)))        if Delta(K)=<7% and (VanillaTV(K)+maximum(Smile(K),0))>=0.001%,        and        G(Delta,        TV)=0.5*(1−exp(−1300*(VanillaTV(K)+maximum(Smile(K),0)))        if Delta(K)=<7% and (VanillaTV(K)+maximum(Smile(K),0))=<0.001%.        In the above exponentials the TV is measured in percentage        units.    -   4. Calculate the bid price and offer price as follows:        -   Bid−Price(K)=max(Price(K)−0.5*Spread(K), min(Price(K),1 bp))            Offer-Price(K)=B id-Price(K)+Spread(K)    -   wherein Price(K) denotes the middle price in basis points (bp)        of the option being priced.    -   5. Find Volatility-Bid and Volatility-Offer that correspond to        Bid-Price(K) and Offer-Price(K). These volatilities are the bid        and offer volatilities.

Reference is now made to FIG. 3, which schematically illustrates asystem for pricing financial derivatives in accordance with anembodiment of the invention. As described in detail above, the systemincludes a database 218 for storing information received from a user200, including details of an option to be priced, as well as real timedata 214, such as market conditions from sources as are known in theart. Market conditions may include, for example, a current spot pricefor the underlying asset (or other value) subject of the option. Theinformation received from the user and the real time market conditionsare processed by an application server 212, which may include anycombination of hardware and/or software known in the art for processingand handling information received from various sources. Applicationserver 212 is preferably associated with a controller, as is known inthe art, which controls and synchronizes the operation of differentparts of the system. Application server 212 is associated with Bid/Offerprocessor 216 which executes the method described above with referenceto FIGS. 2A-2D. The Bid/Offer processor may include any combination ofhardware and/or software known in the art capable of executing thealgorithms described above with reference to FIGS. 2A-2D.

The information from user 200 is preferably received by a web server210, as is known in the art, which is adapted to communicate with aglobal communication network 202, e.g., the Internet, via acommunication modem, as is known in the art. The user may communicatewith web server 210 via the Internet 202 using a personal computer, orany other suitable user interface having a communication modem forestablishing connection with the Internet 202, as is known in the art.In other embodiments of the invention, user 200 may communicate with webserver 202 directly, for example, using a direct telephone connection ora secure socket layer (SSL) connection, as is known in the art. In analternative embodiment of the invention, user 200 is connected directlyto application server 212, for example, via a local area network (LAN),or via any other local communication network known in the art.

The real time data 214 may be received directly by application server212 using any direct connection means as are known in the art.Alternatively, the real time data may be received from sources availableon the global computer communication network, using web server 210.After computing a bid price and an offer price for the option requestedby the user, application server 212 may communicate the computedbid/offer prices to user 200 via web-server 210, as is known in the art,in a format convenient for presentation to the user.

Table 1 below shows three examples demonstrating application of apreferred embodiment of the invention, as described above with referenceto FIGS. 2A-2D. These examples are based on information taken fromforeign exchange exotic option brokers on three different dates. Thedates appear on the trade date row of the table. The brokers provide thematurity date of the option from which the number of days to maturity ismeasured. The remaining inputs include the details of the option, forexample, option parameters such as strike, put or call, barrier, as wellas market conditions relevant to the trade, e.g., spot, forward, ATMvolatility, 25 delta RR, 25 delta butterfly, current interest rates, theTV of the exotic option, and the TV of the corresponding Vanilla option.Actual bid and offer prices for Table 1 are taken from a number ofmarket makers (brokers). The “fair price” entries in Table 1 representthe average of the bid/offer prices presented by the different marketmakers. The fair market price represents the market price of the option.Finally, the bid and offer prices calculated in accordance with apreferred embodiment of the invention are presented at the bottom of thetable, denoted “model price”. These prices are generated from theadjusted mid-market prices and the bid/offer spreads as described above.It is evident from Table 1 that the model of the present inventionprovides a correct bid/offer spread and a correct adjusted mid-marketprice for the exemplary options computed. It is also evident from Table1 that the TV calculated based on the Black-Scholes model does not yieldcorrect results.

Referring to Example 1 in Table 1, the option traded on Feb. 12, 1999,has an expiration date of Jun. 14, 1999, i.e., 122 days after trading.Table 1 also presents additional details of the option as discussedabove. For example, the spot price for the underlying asset of theoption of Example 1 is 114.40, the volatility for this option is 17.35,the forward for this option is −1.86, the theoretical value (TV)calculated based on the Black-Scholes model is 0.38, and thecorresponding Vanilla option price and bid/offer spread are 2.7 and0.25, respectively. Table 1 also presents bid and offer prices for theoption of Example 1, provided by 6 different market makers, which may beaveraged to yield “fair” bid and offer prices of 0.38 and 0.64,respectively. As further shown in Table 1, the bid and offer “model”prices for the option of Example 1 are 0.38 and 0.64, respectively,which values are identical to the “fair” bid and offer prices,respectively. From analyzing Table 1, one skilled in the art willconclude that the results obtained from the model of the presentinvention for the option of Example 1 are remarkably close (essentiallyidentical) to the average fair prices of the same option. Similarly,from analyzing the data of Examples 2 and 3 in Table 1, one skilled inthe art will appreciate that the “model prices” computed in accordancewith the invention for the options of Examples 2 and 3 are substantiallyidentical to the average “fair prices” of these options.

TABLE 1 Example 1 Example 2 Example 3 Interest Rate 1 6.19% 3.10% 6.09%Interest Rate 2 0.19% 6.18% 0.17% Currency 1 U.S dollar Eur U.S. dollarCurrency 2 Japan yen U.S. dollar Japan yen Trade Date 12-Feb-9918-Jan-00 15-Jun-99 Expiration Date 14-Jun-99 16-Nov-00 15-Dec-99 Monthsto Expiration 4 10 6 Days to Expiration 122 303 183 Spot 114.40 1.01120.35 Volatility 17.35 10.75 12.5 Forward −1.86 0.006 −3.15 Strike116.00 1.00 115.00 Put/Call Call Call Put Barrier (trigger) 126.00 1.10100.00 25 Delta Risk −0.375 −0.4 0.1 Reversal 25 Delta Butterfly 0.750.25 0.55 Class RKO RKO RKO TV (exotic) 0.38 0.655 1.64 TV (Vanilla) 2.75.47 2.565 Vanilla Bid/Offer 0.25 0.25 0.25 spread (Vol) Number ofprices 6 5 5 6 market makers Bid Offer Bid Offer Bid Offer 1 0.38 0.7073 99 129 154 2 0.38 0.65 75 91 124 151.5 3 0.36 0.66 75 95 122 152 40.38 0.58 73 98 120 150 5 0.40 0.60 76 89 125 150 6 0.39 0.64 Fair Price(Bid/Offer) 0.38 0.64 .75 .94 1.24 1.51 Model Price 0.38 0.64 .75 .941.24 1.51 (Bid/Offer)

It should be appreciated that the benefit to financial markets fromhaving an accurate model to price derivatives are enormous. First, theaccurate model of the invention enables less experienced users ofderivatives to price them accurately. This advantage alone is importantbecause dealing at wrong prices causes substantial financial losses.Second, by virtue of having an accurate pricing model, the derivativesmarket is expected to become more liquid. Not being able to determinethe correct price of derivatives creates strong dependency on marketmakers and causes users to refrain from using derivatives. By increasingthe use of derivatives, corporations and other hedgers can better hedgetheir cash flows. Third, currently, many corporations and funds, forexample, cannot establish credit lines vis-à-vis each other and arerequired to deal only with banks. Setting up credit lines is acomplicated task and requires a lot of information about the companyrequesting the credit line. Many small companies, for example, cannotestablish credit lines and are thus prevented from using derivatives fortheir hedges. By having an accurate model for market prices any twoparties can deal with each other on a margin basis, even if they do nothave mutual credit lines. For example, such parties can set a margin onthe price of the derivative, as is used in exchanges, that includes thecurrent loss for the seller relative to the current market price. Thusif one of the counter parties defaults during the life of the option,the other one can unwind the derivative with another party at no loss.

The following example illustrates an exemplary margin system. Assumingthat the value calculated by the model of the present invention for themarket price of a given derivative at a given time, t, is P(t). Theprice P(t) may be the middle between the bid price and the offer price,or the offer price, or any other agreed upon function of the bid priceand/or the offer price. The premium paid at the execution of the deal isP(0). The margin amount, in money terms, is denoted M. In this example,at time, t, the seller of the derivative must have in his margin accountfor the buyer of the derivative the following amount:

M(t)=max(0,P(t)−P(0))+X

wherein X can be any positive amount on which the two sides agree uponin advance and is designed to protect the buyer against abrupt moves inthe market. X can also be proportional to the bid/offer price of theoption, at time t. If at some point in time the margin account reacheszero, then the buyer of the derivative may have the right to closehis/her position with the seller and buy the option from a third party.

As a result, in the above example, the market will be open to many moreusers, e.g., corporations, funds, private investors, and otherderivatives users, who can save considerable amounts of money comparedto their situation today. For example, such users would be able to dealat better prices with a variety of corporations and would not be limitedto deal only with the community of market maker. e.g., banks orspecialized brokers, with whom they have succeeded to establish creditlines. It should be appreciated that such development would increase theliquidity of derivatives dramatically.

While the embodiments of the invention shown and described herein arefully capable of achieving the results desired, it is to be understoodthat these embodiments have been shown and described for purposes ofillustration only and not for purposes of limitation. Other variationsin the form and details that occur to those skilled in the art and thatare within the spirit and scope of the invention are not specificallyaddressed. Therefore, the invention is limited only by the appendedclaims.

1-52. (canceled)
 53. A computer-based method of on-line execution ofoption transactions, the method comprising: receiving, by a computingdevice, first data corresponding to at least one parameter defining anoption on an underlying asset; receiving, by the computing device,second data including real-time data of at least one real-time currentmarket condition relating to said underlying asset; determining, by thecomputing device, a bid price and an offer price corresponding to saidoption by setting a bid/offer spread between the bid price and the offerprice based on the first and second data; and executing, by thecomputing device, at least one on-line transaction of the option usingat least one price selected from the group consisting of the bid priceand the offer price.
 54. The method of claim 53, wherein executing saidtransaction comprises selling said option at said offer price.
 55. Themethod of claim 53, wherein executing said transaction comprises buyingsaid option at said bid price.
 56. The method of claim 53 comprisingexecuting said transaction via a communication network.
 57. The methodof claim 53, wherein said first data comprises an indication of at leastone element selected from the group consisting of a type of said option,an expiration date of said option, a trigger for said option, and astrike of said option.
 58. The method of claim 53, wherein said seconddata comprises an indication of at least one element selected from thegroup consisting of a spot value, an interest rate, a volatility, anat-the-money volatility, a 25 delta risk reversal, a 25 delta butterfly,and a 25 delta strangle.
 59. The method of claim 53 comprising computingsaid bid/offer spread based on a corrected theoretical value of saidoption.
 60. The method of claim 59, wherein computing said bid/offerspread comprises: computing a base value for the bid/offer spread usingat least part of said first and second data; and computing saidbid/offer spread by correcting said base value using at least part ofsaid first and second data.
 61. The method of claim 53, wherein saidunderlying asset comprises a financial asset.
 62. The method of claim61, wherein said option is a foreign exchange (FX) option, a Vanillaoption, an option-like financial derivative, or an exotic option. 63.The method of claim 53, wherein said underlying asset is related to atleast one of a commodity, a stock, a bond, an interest rate, and aweather.
 64. A system of on-line execution of option transactions, thesystem comprising: a server to receive via a communication network firstdata corresponding to at least one parameter defining an option on anunderlying asset, and second data including real-time data of at leastone real-time current market condition relating to said underlyingasset; and a processor to determine in real time a bid price and anoffer price corresponding to said option by setting a bid/offer spreadbetween the bid price and the offer price based on the first and seconddata, wherein said server is to execute over the communication networkat least one on-line transaction of the option using at least one priceselected from the group consisting of the bid price and the offer price.65. The system of claim 64, wherein said server is to execute saidtransaction by selling said option at said offer price.
 66. The systemof claim 64, wherein said server is to execute said transaction bybuying said option at said bid price.
 67. The system of claim 64,wherein said first data comprises an indication of at least one elementselected from the group consisting of a type of said option, anexpiration date of said option, a trigger for said option, and a strikeof said option.
 68. The system of claim 64, wherein said second datacomprises an indication of at least one element selected from the groupconsisting of a spot value, an interest rate, a volatility, anat-the-money volatility, a 25 delta risk reversal, a 25 delta butterfly,and a 25 delta strangle.
 69. The system of claim 64, wherein saidprocessor is to determine said bid/offer spread based on a correctedtheoretical value of said option.
 70. The system of claim 69, whereinsaid processor is to determine a base value for the bid/offer spreadusing at least part of said first and second data; and to determine saidbid/offer spread by correcting said base value using at least part ofsaid first and second data.
 71. The system of claim 64, wherein saidunderlying asset comprises a financial asset.
 72. The system of claim71, wherein said option is a foreign exchange (FX) option, a Vanillaoption, an option-like financial derivative, or an exotic option. 73.The system of claim 64, wherein said underlying asset is related to atleast one of a commodity, a stock, a bond, an interest rate, and aweather.